\chapter{Conclusions}
\begin{quote}
Do not seek for information of which you cannot make use.\\
 --Anna C. Brackett
\end{quote}
\section{Major results}
We have presented three major sets of results that have advanced the
field of quantum state characterization.  We have extended the techniques of quantum
state tomography to systems of indistinguishable
particles.  In the process we have been able to quantify the amount of
information available to be measured in a given system of
experimentally indistinguishable particles and used it to put a bound
on the degree of distinguishability in the system.  We have
demonstrated the technique by characterizing states of two and three
photons in the laboratory, and it is to be hoped that this technique
will be adopted in future research into states of indistinguishable
particles.  

We have examined how to improve and optimize ordinary two-qubit
quantum state tomography by trying to match the tomography procedures
to the natural symmetries of the Hilbert space of states.  The
protocol we introduce minimizes redundant information collected and is
therefore able to obtain a better estimate of the quantum state from a
given number of copies than any other method implemented to date.  We
have demonstrated this advantage by analyzing three different quantum
states and found that our method offers a real-world advantage, even
given our experimental limitations.  Additionally, the information collected in this
new tomography procedure can be arranged to form a very natural
description of the quantum state in terms of a discrete Wigner
function.  This function is real-valued and has a connection to classical
probability distributions that is absent for the density matrix.  As a
result, it offers an appealing description of the quantum state that
may play a significant role in quantum information research.

Finally, we have demonstrated that it is occasionally possible to do
without quantum state tomography when one is interested in measuring specific
property of the system, even when that property is not
an observable on the Hilbert space of states.  By measuring the
purity of a single-photon polarization state directly, without first
measuring the density matrix, we have demonstrated a technique that
will prove useful when the exponential dependence of the number
of density matrix elements on the number of particles in the system
becomes too onerous to be practically measurable.  In the process we highlight the fundamental
differences of different methods of preparing mixed polarization states and
point out the necessity of removing entropy from the system rather
than hiding it away in another mode.  The source of randomness
that leads to impurity is an important consideration to keep in mind when designing linear optical quantum
information systems and may play a role in other systems where
particles storing information have many degrees of freedom of which
only one is used.

\section{Future outlook}
With dozens of proposals for physical
systems as candidates for a quantum computing architecture, the
importance of demonstrating high quality entanglement and adequate state
control has become essential to advancing research into any particular
one.  Tomography is now regarded as the gold standard for
proving that a particular quantum state has been made or a given
level of entanglement achieved.  As a result, interest in quantum
state estimation technologies is blossoming, as can
be seen by the proliferation of papers and conferences devoted to
the topic.  The importance of properly understanding quantum state tomography
was made clear when it was pointed out that a Nature
paper\cite{Stevenson2006} describing a
quantum dot photon source had incorrectly inferred that the source was
entangled based on a misreading of tomographic
data\cite{Gilchrist2007}.  As tomography methods become part of the
standard toolkit for quantum information research, the techniques of
quantum state tomography, including those presented in this thesis,
will become day-to-day operations in laboratories everywhere.

At the same time, quantum state estimation is becoming an
important branch of research in its own right.  By examining
closely how information is extracted from quantum mechanical
systems we can begin to probe the strange and delicate interplay
between certainty and randomness that underlies quantum
information.  In this way, little by little, we are beginning to
understand what quantum states really \emph{are}, learning the
limits and uses of quantum information and understanding, as
Rolf Landauer first emphasized, that information is a physical
quantity, as much tied to the physical world as gravity and
angular momentum.  We are discovering that there is a deep
connection between what Nature will allow us to learn about a
system through measurement and the fundamental laws of physics
encoded in the postulates of quantum mechanics\cite{Clifton2003}.  This, arguably more than
any other discovery of the past fifty years, will be what drives
our quest for deeper understanding of the physical world in the
twenty-first century.
